Sharp Quantitative Matrix-Weighted Bounds for Fractional Integrals and Sobolev Inequalities
Abstract
For α∈(0,n), p∈(1,n/α), q:=np/(n−αp), and W∈𝒜p,q, we obtain the following matrix-weighted boundedness of the fractional integral Iα: for any f⃗∈ Lp(Wp), |Iαf⃗|Lq(Wq) ≲[W]𝒜p,q (1−α/n)max{1,p'/q} |f⃗|Lp(Wp), where the implicit positive constant is independent of W and f⃗. For integer n≥2, p∈[1,n), q:=np/(n−p), and W∈𝒜p,q, we establish the following matrix-weighted Sobolev inequality: for any smooth ℂd-valued function f⃗ with compact support, |f⃗|Lq(Wq) ≲[W]𝒜p,q(n−1)/n |WDf⃗|Lp(ℝn,ℂd× n), where the implicit positive constant is independent of W and f⃗ and where Df⃗ is the Jacobian matrix of f⃗. Surprisingly, in both bounds above the exponents of [W]𝒜p,q coincide with the corresponding known optimal scalar exponents, and hence are optimal, which is different from the Calderón--Zygmund operator case. A key idea to obtain the above matrix-weighted boundedness of the fractional integrals is to establish the off-diagonal chain-packing principle of dyadic cubes, which is inspired by a recent work of A. K. Lerner.
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BibTeX
@misc{yang2026sharp,
title = {{Sharp Quantitative Matrix-Weighted Bounds for Fractional Integrals and Sobolev Inequalities}},
author = {Dachun Yang and Wen Yuan and Mingdong Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.34403},
eprint = {2609.34403},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.34403},
}
RIS
TY - UNPB
TI - Sharp Quantitative Matrix-Weighted Bounds for Fractional Integrals and Sobolev Inequalities
AU - Dachun Yang
AU - Wen Yuan
AU - Mingdong Zhang
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2609.34403
UR - https://arxiv.org/abs/2609.34403
AB - For α∈(0,n), p∈(1,n/α), q:=np/(n−αp), and W∈𝒜_p,q, we obtain the following matrix-weighted boundedness of the fractional integral I_α: for any f⃗∈ L^p(W^p), |I_αf⃗|_L^q(W^q) ≲[W]_𝒜_p,q^ (1−α/n)max{1,p'/q} |f⃗|_L^p(W^p), where the implicit positive constant is independent of W and f⃗. For integer n≥2, p∈[1,n), q:=np/(n−p), and W∈𝒜_p,q, we establish the following matrix-weighted Sobolev inequality: for any smooth ℂ^d-valued function f⃗ with compact support, |f⃗|_L^q(W^q) ≲[W]_𝒜_p,q^(n−1)/n |WDf⃗|_L^p(ℝ^n,ℂ^d× n), where the implicit positive constant is independent of W and f⃗ and where Df⃗ is the Jacobian matrix of f⃗. Surprisingly, in both bounds above the exponents of [W]_𝒜_p,q coincide with the corresponding known optimal scalar exponents, and hence are optimal, which is different from the Calderón--Zygmund operator case. A key idea to obtain the above matrix-weighted boundedness of the fractional integrals is to establish the off-diagonal chain-packing principle of dyadic cubes, which is inspired by a recent work of A. K. Lerner.
ER -